Sum of Angles of a Triangle in Non-Euclidean Geometry

Date: 08/01/2001 at 18:54:21
From: Roger
Subject: Three-dimensional figures
Recently I was trying to teach my son that the sum of all the angles
of a triangle is equal to 180 degrees. My son came up with a drawing
of a triangle on a surface of a sphere in which the sides, due to the
shape of the sphere, looked like an arch instead of a straight line
of a normal triangle drawn on a plane surface. When we added the
angles of that particular triangle the sum was definitely greater than
180 degrees. Is that possible? Why?

Date: 08/01/2001 at 23:11:47
From: Doctor Peterson
Subject: Re: Three-dimensional figures
Hi, Roger.
The theorem about the sum of angles in a triangle is from PLANE
geometry, and depends on the parallel postulate, which says that,
given a line and a point, there is exactly one line through the point
parallel to the line. If you move to SPHERICAL geometry, this
postulate is no longer true, and you have a "non-Euclidean" geometry.
(You have to redefine "lines" to be great circles.) Here, the sum of
angles is always greater than 180 degrees; in fact, it turns out that
the excess over 180 degrees is proportional to the area of the
triangle. Try making a triangle containing 1/8 of a sphere, using the
equator and the 0 and 90 degree longitude lines; then try some other
similar triangles to see that this is true.
You can read about this in the Dr. Math archives:
Drawing Triangles
http://mathforum.org/dr.math/problems/joe6.18.97.html
A Triangle with Three Right Angles
http://mathforum.org/dr.math/problems/stine.12.1.99.html
- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/